in \(\mathbb {P}_0^{(n)}\) -probability for \(p=2,\infty \) . Consider the ball \(\mathcal {B}_{n,p}(\epsilon _n)=\lbrace f: \Vert f-f_0\Vert _p<\epsilon _n\rbrace \)  of the metric space \((L^2_{p_X}(\mathcal {X}), \Vert \cdot \Vert _2)\)  if \(p = 2\)  and \((L^\infty (\mathcal {X}), \Vert \cdot \Vert _\infty )\)  if \(p = \infty \) . Note that \(\Vert \cdot \Vert _p\)  is convex and uniformly bounded on the ball \(\mathcal {B}_{n,p}(\epsilon _n)\) . By Theorem 8.8 in [1]}, we have
\(\Vert \hat{f}_n-f_0\Vert _p\le M_n\epsilon _n+\Vert d\Vert _\infty ^{1/2}\Pi _n(\Vert f-f_0\Vert _p>M_n\epsilon _n\mid \mathbb {D}_n),\) 
